Then you're merely in a chicken and egg situation, which can still get very complicated for genetics. For example, most prion diseases are also "environmental", yet they've surely started somewhere spontaneously. Similarly, viruses make for many mutations, some of which stick around in species' genomes.

The people who end up being incompatible with the biology their genes produce almost always end up dead at young ages, even if they make it past birth. The people who find themselves in the middle ground of barely surviving their own bodies are truly often unique, interesting medical cases. Like the bone disease that rapidly calcifies injuries or other stressed tissue. If you read about the disease in some flowery old language, you'd think it was just a high tale.

Given the myriad of diseases like that, I'd be shocked if there were no saliva related ones (kinda like Sjogren disease...) for cats. Biology is insane.

It's usually a better question of how rare something is more than if it's possible. Especially if it's something as basic as allergies. People can be allergic to their own sweat for crying out loud. I'd be shocked if it never happened for cats.